MODULATION CLASSIFICATION IN FADING CHANNELS USING ANTENNA ARRAYS Ali Abdi1, Octavia A. Dobre1, Rahul Choudhry1, Yeheskel Bar-Ness1, and Wei Su2 1CCSPR, Dept. of ECE, New Jersey Institute of Technology, Newark, NJ 07102, USA 2 CECOM, Fort Monmouth, NJ 07703, USA ABSTRACT ϕ0 is the channel phase (which also includes the carrier ()i N phase offset), {}skk=1 represents the N complex Blind modulation classification (MC) is an intermediate transmitted data symbols, taken from the i-th finite- step between signal detection and demodulation, and plays a alphabet modulation format, and † is the transpose key role in various civilian and military applications In this operator. Depending on the classification approach, in paper, first we provide an overview of decision-theoretic Section IV we consider α0 and ϕ0 as random variables, MC approaches. Then we derive the average likelihood ratio denoted by α and ϕ , respectively, and average over (ALR) based classifier for linear and nonlinear them, whereas in Section V, α and ϕ are considered as modulations, in noisy channels with unknown carrier phase 0 0 unknown deterministic quantities, for which αˆ0 and ϕˆ0 offset and also in Rayleigh fading channels. Since these are provided as their estimates. In any case, we treat the ALR-based classifiers are complex to implement, we then ()i N data symbols {}skk=1 as independent and identically develop a quasi hybrid likelihood ratio (QHLR) based distributed (iid) randoms, and average over them. With classifier, where the unknown parameters are estimated symbol period T0 , rectangular unit-amplitude pulse shape using low-complexity techniques. This QHLR-based ut() of length T , and j 2 =−1 , st(;u ) is given by T0 0 i,0 classifier is much simpler to implement and is also N jϕ0 ()i applicable to any fading distribution, including Rayleigh and st(;uikT,0 )=−−≤≤α 0 e s () tu ( t ( k 1) T 0 ),0 t NT 0 . (1) Rice. Afterwards, we propose a generic multi-antenna ∑k =1 0 classifier for linear and nonlinear modulations, using an Eq. (1) is applicable to M-ary ASK, PSK, QAM, and antenna array at the receiver. This classifier has the FSK modulations. In MASK we have potential to improve the performance of traditional single- sM(MASK) ∈ {±± 1, 3,..., ± ( − 1)} , where M is even. The antenna classifiers, including the proposed QHLR-based symbols of MPSK are given by se(MPSK) = jθm , in which algorithm, via spatial diversity. Simulation results are θm = 2π mM , m=− 0, 1, ..., M 1 , with M as a power of provided to show the performance enhancement offered by two. For square and cross QAM symbols, see [1]. In the new QHLR-based multi-antenna classifier, in a variety MFSK, the symbols are given by ste(MFSK)()= jft 2π m , such of channel and fading conditions. that fmd= ±±ff00, 3 d , ..., ± ( Mf − 1) d 0, with fd 0 as the frequency spacing between any two adjacent constellation I. INTRODUCTION points and M as a power of two. As mentioned in the Modulation identification of a received signal is of footnote of this page, for linear modulations, the data importance in a variety of military and commercial symbols do not depend on t. This property simplifies the applications. Some examples include spectrum monitoring general classifier derived in Section IV. and management, surveillance and control of broadcasting Let us define the variance of the zero-mean i-th activities, adaptive transmission schemes, and electronic ()i ()i 2 constellation as ϑ = Es[|k | ] , iN= 1,2,..., mod , where E[.] warfare. Our comprehensive literature survey [3] shows denotes the mathematical expectation. Then the signal that although this topic has been extensively studied, less power, defined by attention has been paid to modulation classification in NT0 fading channels, and also the utilization of the spatial ()i −12 SNTEstdt00= () [|(;)|]ui ,0, (2) diversity, provided by antenna arrays, in such channels. In ∫0 ()i 2()i this paper, we study these two topics in a systematic way. can be shown to be S00= α ϑ , after substituting (1) into (2). For equiprobable Mi constellation points of the i-th II. SIGNAL, NOISE, AND CHANNEL MODELS ()ii− 1Mi ()2 modulation, obviously one has ϑ = Msim∑m=1||. Note Let st(;u ) represent the noise-free baseband complex ()i 2 i,0 that for PSK and FSK ||1sm = , mM= 1,2,..., i , whereas envelope of the received signal, coming from the i-th ()i 2 in ASK and QAM, ||sm takes different values. modulation format, iN= 1,2,..., , where N is the mod mod In the presence of noise, for the baseband received number of candidates modulations that we are looking at. complex envelope we have The vector ui,0 , which corresponds to the i-th modulation format, denotes the vector of unknown quantities at the rt()= st (;ui,0 )+≤≤ nt (),0 t NT 0 , (3) receiver. In this paper we consider a frequency-flat slowly- where nt() is the complex additive white Gaussian noise varying multipath fading channel with (AWGN) with two-sided power spectral density N ()i N † 1 0 * uikk,0= [αϕ 0 0 {}]s = 1 , where α0 is the channel amplitude, (W/Hz), and the correlation Entn[() ( t+=τ )] N0δτ (), such ____________________________________________________________ that * is the complex conjugate and δ(.) is Dirac delta. 1 As we discuss in the first paragraph after eq. (1), only for frequency shift keying (FSK), the data symbols depend on t. For others such as amplitude shift keying (ASK), phase shift keying not depend upon t. To simplify the notation, we most often drop ()i (PSK), and quadrature amplitude modulation (QAM), sk ’s do this t-dependence, unless otherwise mentioned. 1 of 7 III. DECISION THEORY AND LIKELIHOOD search, to find the ML estimates [18], for each hypothesis. To design the modulation classifier, in order to Moreover, for nested constellations such as BPSK/QPSK, and 16-QAM/64-QAM, the likelihood function in GLRT determine what modulation has been received, out of N mod equally likely candidates, we take the decision-theoretic can take the same numerical values, which in turn leads to approach, which is optimal under certain conditions [2]. incorrect classification [18] [26]. This approach, which works based upon the maximum HLRT is a combination of ALRT and GLRT, in an likelihood (ML) principle, requires the likelihood function attempt to avoid the disadvantages of both techniques, of rt() over the interval 0 ≤≤tNT0 . Using the complex while employing their useful properties. In HLRT, the Gaussian distribution of nt() and for the i-th hypothesis unknown data symbols are considered as random variables Hi (the i-th modulation format), the conditional likelihood and are averaged out, whereas the unknown parameters function of rt(), conditioned on the unknown vector ui,0 , are treated as deterministic unknowns, eventually replaced can be shown to be [2] [5] by their ML estimates. To write the likelihood function, we Ξ=[()|rtu , H ] break ui,0 to two vectors w0 and si such that ii,0 ††† uwsii,0= [] 0 . The vector w0 contains the unknown NT NT † 21002 parameters, whereas s = [...]ss s represents the exp Rer ( t ) s* ( t ; ) dt s ( t ; ) dt , iN12 uuii,0− ,0 (4) unknown data symbols. With this notation, the likelihood NN∫∫00 00 function, conditioned on w , is where Re[.] gives the real part. To derive the likelihood 0 ()i ()i function of rt() from (4), Ξ [()]rt , three techniques are Ξ=Ξ[()|rtwwsss00 ] [()| rt ,ii , H ]( p i| H ii ) d . (9) proposed in the literature, that we discuss in the sequel. ∫ ()i With the ML estimate of w0 for each candidate Once Ξ [()]rt is calculated for all the possible N mod candidate modulations, based on the observed rt() over modulation 0 ≤≤tNT0 , one can make the decision according to ML() i wwˆ i,0 = arg maxΞ [rt ( ) | ] , (10) Choosei as the received modulation w the HLRT likelihood function is eventually given by ifirt=Ξ arg max()i [ ( )]. (5) ()i ()iML 1≤≤iN mod Ξ=ΞHi[()]rt [()| rt wˆ ,0 ]. (11) Obviously our modulation classification is a multiple Averaging over the data symbols in HLRT removes the composite hypothesis testing problem, due to the unknown nested constellations problem of GLRT. However, finding ()i data symbols {}sk , as well as the unknown parameters, the ML estimates of the unknown parameters in HLRT which are α0 and ϕ0 in this paper. Based on our still entails an exhaustive search, which makes its comprehensive literature survey [3] [4], three methods are implementation complex. proposed so far, to handle the unknown quantities: average In this paper, first we derive, in Section IV, closed-form likelihood ratio test (ALRT) [6]-[18], generalized likelihood expressions for the ALRT-based likelihood function in ratio test (GLRT) [19], [20], and hybrid likelihood ratio AWGN and Rayleigh fading channels, for any kind of test (HLRT) [19], [21]-[25]. memoryless modulation, including ASK, PSK, QAM, FSK, The unknown quantities in ALRT are considered as etc. Due to the exponential complexity of these ALRT- random variables, with a certain joint probability density based classifiers, as well as not being applicable to other function (PDF), pH(|uii ), and the likelihood function is types of fading such as Rice, Weibull, Nakagami [1], we derived by averaging the conditional likelihood function then propose a quasi HLRT (QHLRT) approach in Section with respect to it V. In QHLRT, we use non-ML parameter estimators, which are simple yet accurate enough to provide a good Ξ=Ξ()i [()]rt [()| rtuu|u , H ]( p H ) d . (6) Aiiiii∫ classification performance, with much less computational complexity, also applicable to any fading distribution. If the chosen pH(|uii ) is the same as the true PDF, then ALRT is the optimal classifier, i.e., maximizes the Then in Section VI we introduce a multi-antenna classifier, probability of correct classification.